Bevel modifier auto segments
Short answer: No

If all what you want is smooth edges, then you are probably better off using a higher number of segments that's high enough for the critical parts, according to your artist's opinion.

First one should discern an adaptive number of segments for all edges together or for each edge.  

For all edges together:
Calculating the number of segments for a given segment length has no closed mathematical solution and requires numerical approximation. Also some bevels don't have a constant offset but are smaller on one end than the other. You have to find the biggest segment width and increase the number of segments iteratively until your maximum width/offset is not exceeded.

For each edge individually:
 1. Using different number of segments for each edge does not allow a clean topology in the intersections (the vertices where bevelled edges meet). Unless you use Ngons for all intersections, there is no known good solution.
 2. In the video, multiple bevels are subsequently used. You can clearly see that the intersection is much different from a usual bevel (top face of the cube is a Ngon). If you have 20 edges and each should have a variable number of segments, one can think of 20 subsequent bevel modifiers. The result will depend on the order which edge is bevelled first and so on. 3. The same problem as in "for all edges" applies here as well.

The used formulae in the video is not accurate, it is just a bit correlated. The segment width or offset is far from constant.

Edit:
Approximate answer for 90° bevels with profile 0.5:

import bpy
C = bpy.context
from math import pi, ceil

# The approximate size of the segments
segment_size = 0.05

# The modifier you want to alter
modifier = C.object.modifiers["Bevel"]

# This is the arc length of a 90° corner, which is close to the
# summed length of the segments
arc = pi*modifier.width/8
seg = ceil(arc/segment_size)
if seg &lt; 1:
   seg = 1

bpy.context.object.modifiers["Bevel"].segments = seg


Anything more general will require:


Face angles at every bevelled edge
3 nested numeric search algorithms